Proceedings of the 14th International Heat Transfer Conference IHTC14 August 08-13, 2010, Washington, DC, USA IHTC14-22023 REYNOLDS, MAXWELL AND THE RADIOMETER, REVISITED Holger Martin Thermische Verfahrenstechnik, Karlsruhe Institute of Technology (KIT) Karlsruhe, Germany ABSTRACT later by G. Hettner, goes through a maximum. Albert Einstein’s In 1969, S. G. Brush and C. W. F. Everitt published a approximate solution of the problem happens to give the order historical review, that was reprinted as subchapter 5.5 Maxwell, of magnitude of the forces in the maximum range. Osborne Reynolds, and the radiometer, in Stephen G. Brush’s famous book The Kind of Motion We Call Heat. This review A comprehensive formula and a graph of the these forces covers the history of the explanation of the forces acting on the versus pressure combines all the relevant theories by Knudsen vanes of Crookes radiometer up to the end of the 19th century. (1910), Einstein (1924), Maxwell (1879) (and Hettner (1926), The forces moving the vanes in Crookes radiometer (which are Sexl (1928), and Epstein (1929) who found mathematical not due to radiation pressure, as initially believed by Crookes solutions for Maxwells creeping flow equations for non- and Maxwell) have been recognized as thermal effects of the isothermal spheres and circular discs, which are important for remaining gas by Reynolds – from his experimental and thermophoresis and for the radiometer). theoretical work on Thermal Transpiration and Impulsion, in 1879 – and by the development of the differential equations The mechanism of Thermal Creeping Flow will become of describing Thermal Creeping Flow, induced by tangential increasing interest in micro- and submicro-channels in various stresses due to a temperature gradient on a solid surface by new applications, so it ought to be known to every graduate Maxwell, earlier in the same year, 1879. student of heat transfer in the future. These fundamental physical laws have not yet made their That’s one of the reasons why some authors have recently way into the majority of textbooks of heat transfer and fluid questioned the validity of the classical Navier-Stokes, Fourier, mechanics so far. and Fick equations: Dieter Straub (1996) published a book on an Alternative Mathematical Theory of Non-equilibrium A literature research about the terms of Thermal Phenomena. Howard Brenner (since 2005) wrote a number of Transpiration and Thermal Creeping Flow, in connection with papers, like Navier-Stokes, revisited, and Bi-velocity hydro- the radiometer forces, resulted in a large number of interesting dynamics, explicitly pointing to the forces acting on the vanes papers; not only the original ones as mentioned in subchapter of the lightmill, to thermophoresis and related phenomena. 5.5 of Brush’s book, but many more in the earlier twentieth Franz Durst (since 2006) also developed modifications of the century, by Martin Knudsen, Wilhelm Westphal, Albert classical Navier-Stokes equations. So, Reynolds, Maxwell, and Einstein, Theodor Sexl, Paul Epstein and others. the radiometer may finally have initiated a revision of the fundamental equations of thermofluiddynamics and heat- and The forces as calculated from free molecular flow (by mass transfer. Knudsen), increase linearly with pressure, while the forces from Maxwell’s Thermal Creeping Flow decrease with INTRODUCTION pressure. In an intermediate range of pressures, depending on The history of the theoretical explanations of the forces the characteristic geometrical dimensions of flow channels or acting on the vanes of Crookes’s radiometer (and the related radiometer vanes, an appropriate interpolation between these problems of Thermal Transpiration and Thermophoresis) can two kinds of forces, as suggested by Wilhelm Westphal and 1 Copyright © 2010 by ASME now be subdivided into three separate periods. Each of these 1860, and 1861, these German scientists had already periods had a duration of about 20 to 25 years: discovered the elements cesium and rubidium by their new 1873 – 1897: The period starting with Crookes’s discovery method. Crookes tried to find out the molecular mass of his of the existence of a radiometric force. Important papers were new element by a series of careful weighings, partly under written by Crookes (1875 – 1877), by Reynolds (1876 – 1879), vacuum. During these experiments he observed that a hot body and by Maxwell (1879). This first period is very well covered in the vicinity of a mass on the balance can change the weight by chapter 5.5 of Brush’s book on the history of the kinetic of that mass! When put below the mass on the balance the theory of gases during the 19th century. About 20 authors weight decreased, while it increased when the hot body was put published at least 30 papers on the topic from 1874 to 1897. above it. His literature studies showed him that Fresnel had 1909 – 1930: This second period of research is characte- found this same effect already in 1825. But when Crookes rized by Knudsen’s work (1909–1930), by the experimental observed it, around 1873, James Clerk Maxwell had already work of Westphal (1910–19..) and Hettler (1924–1928), as postulated, and theoretically predicted the existence of a well as by theoretical work by Einstein (1924), Hettler (1926– pressure exerted by radiation. Crookes built several 1928), Epstein (1927), and Sexl (1928). radiometers, typically with four circular (or quadratic) discs as 1980 – 2010: New interest in the problem came up for vanes mounted on a vertical axis which could rotate around this several reasons after a period of about 50 years of relative axis on a needle in an evacuated glass bulb. The vanes were scarce interest in the late 20th and the beginning 21st centuries. made from pith (and other materials), blackened by soot on one Microchannel applications made transitional effects, belonging side and reflecting on the other side (pith is white by nature). to Knudsen numbers of the order of one more important. That’s why new patents on so-called Knudsen pumps, based on Figure 1 Crookes radiometer thermal transpiration, and, consequently, modifications of the Navier-Stokes, Fourier, and Fick equations were proposed The glass bulb was evacuated as good since 1996 by Straub, since 2005 by Brenner, and since 2006 as possible at the time, and then sealed. by Durst. Sunlight, or the light from a candle, or a number of candles, caused the vanes to NOMENCLATURE rotate quickly around their axis, which d m diameter (tube, disc) was demonstrated by Crookes at a F N force meeting of the Royal Society. This L m length, L2 = (π/4)d2 area demonstration of a surprising force ~ M kg kmol-1 molar mass caused a hype of interest in the n kmol m-2 s-1 molar flux radiometer problem, which led to a p Pa pressure great number of papers published on r m radius this topic within the next few years. ~ ~ This first part of the story is well R J kg-1K-1 gas constant (= R / M ) ~ documented by subchapter 5.5 of S. G. R J kmol-1 K-1 universal gas constant Brush’s book The Kind of Motion We t s time Call Heat1. Crookes’s original idea, T K temperature namely that the radiometer were rotating because of a direct u m s-1 velocity in x-direction effect of radiation pressure, was soon shown to be wrong by v m s-1 velocity in y-direction other scientists. w m s-1 velocity in z-direction At least three reasons were found why it couldn’t be the direct effect of radiation: If it where the light pressure the Δ 1 difference rotation must have been with the reflecting surfaces being Λ m mean free path -3 pushed away from the radiation, which is just opposite to the ρ kg m density observed motion. The order of magnitude of the light pressure η Pas viscosity (as calculated from Maxwell’s theory) was much lower than the forces observed. And finally, if it were the direct effect of radiation, the bulb, when suspended on a thin wire must show a CROOKES, REYNOLDS, MAXWELL tendency to rotate in the same direction as the vanes (due to the The starting point of the whole story may be seen in the friction of the needle bearing). In 1876, Arthur Schuster, a discovery of a new kind of forces observed by the chemist younger colleague of Reynolds at Owens College in William Crookes, around 1873, in his private laboratory in Manchester showed that the bulb rotates in the direction London. In 1861, he had discovered a new element, thallium, opposite to the vanes. From this experimental result it was easy by using the spectral analysis, developed by the chemist to conclude that the remaining gas in the (partly) evacuated Bunsen and the physicist Kirchhoff in 1859 at Heidelberg. In bulb must play a crucial role in the motion of the vanes. 2 Copyright © 2010 by ASME Osborne Reynolds, the first professor of engineering at Owens For higher pressures, however, this simple law, derived from College in Manchester, started a fundamental research program the kinetic theory, doesn’t apply any more. Reynolds’s thermal to find out the nature of the forces acting on the vanes of the transpiration experiments show that the pressure difference lightmill (Crookes’s radiometer). Schuster’s and Reynolds’s goes through a maximum and tends to decrease inversely with papers about their research concerning the radiometer were pressure for higher pressures. Figure 2 shows Reynolds’s data published in 1876. A number of other authors discussed the for stucco slabs with two different pore sizes and experiments 1 topic in the period from 1874 to 1897 (see Brush ). Reynolds with air and hydrogen replotted as Δp/pmax versus p/popt, where tried to solve the problem by careful experiments.